CN110322938B - Determination and search method for topological electronic material - Google Patents
Determination and search method for topological electronic material Download PDFInfo
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Abstract
The invention provides a method for judging a topological electronic material, which comprises the following steps: step 1: judging the magnetism and the metallicity of the given material, and selecting a non-magnetic and non-metallic material; step 2: calculating an electron wave function and an energy eigenvalue of a high symmetry point in the Brillouin zone based on the structure file of the selected material; and step 3: and (4) judging whether degeneracy exists at the energy band occupied by the material at the high-symmetry point or not based on the electronic wave function and the energy eigenvalue obtained in the step (2). The method has simple process, and can comprehensively and quickly find out all topological materials which can be judged by using the symmetrical eigenvalues.
Description
Technical Field
The invention relates to the field of computational physics and material science, in particular to a method for judging and searching a topological electronic material.
Background
Topological electronic materials are a new class of materials discovered in the physics of condensed states in recent years, including topological insulator materials and topological semimetals. The method is mainly characterized by having non-trivial topological properties and being embodied in two aspects: 1. on the surface of the topological insulator material, surface electronic states (such as electronic states without back scattering and the like) which cannot be realized in a general two-dimensional material exist; 2. in the bulk of topological semi-metals, there is a phenomenon called "quantum anomaly" such as linear negative magnetoresistance. Due to the characteristics, the topological electronic material has application prospects in various aspects such as high fault-tolerant topological quantum computing, ultralow energy consumption electronic devices and the like. Many computational physicists and material scientists are also actively looking for newer and better materials of this type.
However, it is not easy to find new topological electronic materials. The existing method for judging the topological electronic material needs to firstly perform first principle calculation on Brillouin zone dense acquisition points (generally dozens of or hundreds of points), then calculate 'topological invariant', and judge whether the material is topological or not according to the numerical value of the topological invariant. Thus, for a given compound, it is necessary to calculate its "topological invariants" in order to determine in advance whether it has topological properties. The topology invariance is a complex mathematical concept firstly, and relates to contemporary mathematical knowledge which is not possessed by scientists in most fields, such as 'algebraic topology'; secondly, even under the condition of complete mathematical knowledge, the implementation of the calculation of the topology invariant in the first principle calculation software is very complicated, and the time cost and the labor cost are very high. Therefore, a simple and easy method for determining and searching for topological electronic materials is urgently needed.
Disclosure of Invention
Therefore, the present invention is directed to overcome the above-mentioned drawbacks of the prior art, and to provide a method for determining a topological electronic material, which is comprehensive, fast and automatic, and comprises the following steps:
step 1: judging the magnetism and the metallicity of the given material, and selecting a non-magnetic and non-metallic material;
step 2: calculating an electron wave function and an energy eigenvalue of a high symmetry point in the Brillouin zone based on the structure file of the selected material;
and step 3: and (4) judging whether degeneracy exists at the energy band occupied by the material at the high-symmetry point or not based on the electronic wave function and the energy eigenvalue obtained in the step (2).
According to the determination method of topological electronic material of the present invention, preferably, the step 2 further comprises performing symmetry check and structure normalization on the structure file of the selected material.
According to the determination method of the topological electronic material, in step 3, preferably, if degeneracy exists at the high-symmetry-point occupied energy band, the material is determined to belong to the high-symmetry-point topological semi-metal.
According to the determination method of the topological electronic material of the present invention, preferably, if there is no degeneracy of the material at the high symmetry point occupied energy band, the following step 4 is performed:
and 4, step 4: and judging whether the energy band of the material meets the compatibility relation.
According to the determination method of the topological electronic material, in step 3, preferably, the direct energy gap of the material is obtained according to the energy eigenvalue, and if the direct energy gap is greater than or equal to 2meV, the material is determined to have no degeneracy at the high-symmetry-point occupied energy band.
According to the determination method of the topological electronic material, in step 3, preferably, a direct energy gap of the material is obtained according to the intrinsic energy value, and if the direct energy gap is less than 2meV, degenerate state calculation is performed.
According to the determination method of the topological electronic material of the present invention, preferably, if the energy band of the material does not satisfy the compatibility relationship, the material belongs to "high symmetry line topological semi-metal".
According to the determination method of the topological electronic material of the present invention, preferably, if the energy band of the material satisfies the compatibility relationship, the following step 5 is performed:
and 5: and calculating the symmetry index of the material.
According to the method for determining a topological electronic material of the present invention, preferably, in step 5, when the spin orbit coupling effect is ignored, if the symmetry index is not all 0, the material is determined to belong to a "common point topological semi-metal".
According to the method for determining a topological electronic material of the present invention, it is preferable that, in step 5, when the spin orbit coupling effect is considered, if the symmetry index is not all 0, the material is determined to be a "topological insulator" or a "topological crystal insulator".
According to the method for determining the topological electronic material, the material is preferably determined to belong to a 'topological insulator' if the last strong topological index in the symmetry indexes is an odd number; and if the last strong topological index in the symmetry indexes is an even number, judging that the material belongs to a topological crystal insulator.
The method provided by the invention can greatly simplify the judgment process of the topological electronic material, and can find out all topological materials which can be judged by using the symmetry eigenvalue through the full-disk scanning of the existing material database.
Drawings
Embodiments of the invention are further described below with reference to the accompanying drawings, in which:
FIG. 1 is a flow chart of a method of determining topological electronic material in accordance with the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be further described in detail by embodiments with reference to the accompanying drawings. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
In general, referring to the flow chart shown in fig. 1, the determination of the topological electronic material of the present invention comprises the steps of:
step 1: judging the magnetism and the metallicity of the material, and selecting a non-magnetic and non-metallic material;
step 2: preferably, the material selected in the step 1 is subjected to symmetry checking and structural normalization;
and step 3: calculating an electron wave function and an energy eigenvalue of the material at a high symmetry point of the Brillouin zone;
and 4, step 4: judging whether degeneracy exists at the position where the material occupies the energy band with high symmetry point, if so, judging the material to be 'high symmetry point topology semimetal', if not, performing the following step 5, because only the material with the high symmetry point direct energy gap smaller than 2meV is possible to be 'high symmetry point topology semimetal', firstly, judging the direct energy gap of the material from the energy eigenvalue obtained in the step 3, if not larger than 2meV, directly judging the non-degeneracy and performing the following step 5, and if not larger than 2meV, performing relevant calculation of degeneracy judgment;
and 5: judging whether the energy band of the material meets the compatibility relation, if not, judging that the material is 'high symmetry line topological semimetal', and if so, performing the following step 6;
step 6: and calculating a symmetry index of the material, and judging whether the material is a 'general point topology semimetal' (when the spin orbit coupling effect is ignored) or a 'topology insulator' or a 'topology crystal insulator' (when the spin orbit coupling effect is considered) according to the symmetry index.
The above steps are explained in detail below with reference to examples of the present invention.
Step 1: giving a material, obtaining The magnetic moment of The material from The Materials Project website, calculating The number of electrons of The material according to The number of each element in The primitive cell, judging The material to be a non-magnetic material if The magnetic moment is less than 0.1 muB, and selecting The non-magnetic and non-metal Materials according to The standard to perform The following steps if The number of electrons of The material is an odd number.
Step 2: usually the symmetry of a material is known, but the structure file of the material for calculation does not strictly have known symmetry due to errors and the like. The structure file is a file that stores the coordinates of the position where each atom in the material is located. In order to ensure the accuracy of the calculation result, the coordinates of the atoms in the material need to be slightly adjusted to have corresponding symmetry. According to one embodiment of the invention, this is done by a Phonopy software package, where the atomic coordinates are moved from 10 during the symmetry determination process-5The angstroms are gradually relaxed to 10-1Angstrom until the correct symmetry is found, if precision is 10-1The structure file for this material is considered erroneous if no symmetry has been found at angstrom. In addition to checking symmetryThe basis vectors of the material primitive cells are normalized, namely the basis vectors of the materials belonging to the same Bravais lattice have the same specification through a similarity transformation. This is required because the matrices of different symmetric operations under different coordinate basis vectors are different, and in order to facilitate the subsequent calculation of the symmetry eigenvalues of the wave function, it is necessary to normalize the basis vectors of the material cells in one embodiment of the present invention, so that the symmetric operation matrices belonging to the same spatial group of bravais lattices are identical.
And step 3: the Vienna Ab initio Simulation Package (VASP) software Package is used for calculating the electronic wave function and the energy eigenvalue of the high symmetry point of the material in the Brillouin zone. According to Song Z, Zhang T, Fang Z, et al, quantitative mapping, between technical schemes and topology in principles [ J ]. Nature communications,2018,9(1):3530. and Song Z, Zhang T, Fang C.Diagnosis for non-quantitative polar correlation in the present invention of space-ordered correlation [ J ]. Physical Review X,2018,8(3):031069. for different space groups, the present invention only needs to calculate certain specific high symmetry points to obtain the topology classification corresponding to the space group. It is known from the above document which high symmetry points need to be calculated for different space groups. When VASP is used for first-order calculation, self-consistent calculation is firstly carried out to generate a charge density file, then the charge density file is read to carry out non-self-consistent energy band calculation, and output files such as energy eigenvalue and wave function are generated for next analysis.
According to one embodiment of the invention, only the electronic wave functions of a few high symmetry points need to be calculated, so that the calculation speed is greatly increased. This calculation is performed separately for the case of spin-orbit coupling (soc-setting) and the case of spin-orbit coupling (nsoc-setting), and all the following steps are also independently calculated for the two cases. The actual materials are all spin-orbit coupled, and in the algorithm, the topological classification given after the spin-orbit coupling is considered corresponds to the real topological classification of the materials. On the other hand, the present algorithm also gives a topological classification when spin-orbit coupling is ignored, which is physically meaningful, although it does not correspond to the true topological classification of the material. Ignoring the topological classification of spin-orbit coupling helps to understand how the topological properties of spin-orbit coupling are considered. In addition, for some elements with weak spin-orbit coupling effect, the topological classification given when the spin-orbit coupling is ignored may be better in accordance with the experiment due to the limited observation precision in the experiment and the like. Therefore, the algorithm of the invention can give two topological classification results when spin orbit coupling is considered and when spin orbit coupling is not considered for one material, so as to provide better reference for related scientific researchers.
And 4, step 4: theoretically, the degenerate energy bands have the same energy eigenvalues, but in actual calculation, the degenerate energy bands have a slight energy difference due to the precision and the like. On the other hand, the irreducible representation which is different for each space group has a definite dimension, and the electronic wave function of the material belonging to a certain space group has the symmetry of the space group, so that each energy band of the material forms a basis function of a certain one-dimensional representation of the space group if not degenerated, and several degenerated energy bands form a basis function of a certain high-dimensional representation if degenerated. According to the property, when determining whether degeneracy exists at the occupied energy band, only a maximum degeneracy error needs to be given, and within the error range, an irreducible representation to which the Nth energy band belongs is calculated. If the degenerate band belonging to the irreducible representation comprises the N-th and N + 1-th bands, then this high symmetry point is degenerate at the occupied number band. If degeneracy exists, the material belongs to high-symmetry point topological semimetal, and if degeneracy does not exist, the next judgment is carried out.
The following describes a calculation method for band irreducible representation according to an embodiment of the present invention:
the first step is as follows: and calculating the symmetry eigenvalue, namely the characteristic mark, of the wave function under the space group symmetry operation. The wave function of the VASP is expanded by a plane wave basis vector, and a plane wave basis vector expansion coefficient of the wave function can be read from the output file WAVECAR of the VASP. Let the wave function at a certain point (set as k point) of the Brillouin zone
Is a radical of some one-dimensional representationFunction of where ei(k+G)rIs the plane wave component with the reciprocal lattice vector G,
is the corresponding plane wave coefficient, then for some symmetric operation R, τ there is R τ ψk(r)＞＝λ|ψk(R)), where λ is the signature of the wave function under the symmetric operation, R is the three-dimensional rotation matrix in the symmetric operation { R, τ } and τ is the translational part of the operation. The formula λ ═ ψ can be used in practical calculationsk(r)|{R|τ}|ψk(r)) (assuming the wave function has been normalized, the mode power of the wave function needs to be removed if it is not). When the spin-orbit coupling effect is considered, the SU (2) matrix corresponding to the symmetric operation is also required to flip the wave function components at spin and at spin, which contributes a phase factor modulo 1 to the signature.
The second step is that: and judging which irreducible basis function of the space group is formed by the wave function according to the characteristic mark of the wave function under each symmetrical operation. In actual calculation, the obtained feature labels and each irreducible representation of the space group are used to make inner products by using the orthogonal theorem of group representation, and the expression that the inner products are 1 after normalization is obtained. In actual calculation, the inner product is not strictly 1 due to the calculation errors such as limited VASP plane wave truncation energy and the like, and the maximum error set by the method is 0.05. If the integral number of representations cannot be obtained within the error range, the process returns to step 3, and the wave function of the material needs to be recalculated with improved accuracy. When the orthogonal theorem is performed, the spatial group feature table under a certain specification is required to be selected as a standard. The algorithm judges irreducible representation by utilizing a double-group irreducible representation characteristic index table on a Bilbo Crystalgraphical Server (BCS), which requires that the selected specification is consistent with the BCS in the calculation process, wherein the selection of primitive cell basic vectors, the selection of primitive cell coordinate origin, the selection of high symmetry point coordinates, the selection of SU (2) matrix corresponding to each three-dimensional rotation matrix and the like are included. The method for taking the original cell coordinate origin determines the method for taking the translation part corresponding to the non-simple operation in the non-simple space group. In addition, it should be noted that double population irreducible on BCS means that time-reversal symmetry is not considered, and is always present because the materials calculated by the present invention are all non-magnetic. Additional degeneracy between irreducible representations of the space groups occurs after consideration of time-reversed symmetry, and specific representation degeneracy can be found in the "dual space group band representation and base band representation" templates of BCS.
The conditions of high-dimensional representation are similar, and when the feature label is solved, only the symmetrical eigenvalues of the degenerate energy bands are needed to be respectively calculated and summed, so that the feature label represented by the high-dimensional irreducible representation can be obtained. It should be noted that if the energy bands belonging to the same high-dimensional representation cannot be represented by a single energy band, the eigenvalues of symmetry of each energy band are added to form the signature of the high-dimensional representation. The algorithm of the invention needs to divide the energy bands into sets of degenerate energy bands and then represents each set. The degeneracy error is set to the smaller of 5meV and 0.1 × gap, where gap is defined as the energy difference between the nth and N +1 bands of the high symmetry point.
And 5: after determining that there is no degeneracy at each highly symmetric point occupied number band, the irreducible representation of the occupied number and its following N bands, where N is the number of electrons of the material, is calculated according to the calculation method of the band irreducible representation mentioned earlier. After the energy band representation of each high symmetry point is calculated, whether the energy band representation of any two high symmetry points meets the compatibility relation on each high symmetry line connecting the two points is judged, if not, the energy band intersection of a conduction band and a valence band exists on the high symmetry line between the two high symmetry points, the two high symmetry lines belong to high symmetry line topology semimetal, and if the relation is met, the next step of judgment is carried out. The irreducible compatibility relationship between each high symmetry point of the space group and each connected high symmetry line can be obtained on the BCS, the compatibility relationship between the two high symmetry points can be obtained by matching the same high symmetry line between the two high symmetry points, if the two high symmetry lines are connected, a plurality of compatibility relationships exist between the two high symmetry points, and the condition that one high symmetry line represents that energy bands are crossed is broken.
Step 6: according to Song Z, Zhang T, Fang Z, et al.Quantitative mapping between topological symmetry and topologic in principles [ J ]. Nature communications,2018,9(1):3530. and Song Z, Zhang T, Fang C.Diagnosis for non-macromolecular polar chemistry in the present of spin-ordered linking [ J ]. Physical Review X,2018,8(3):031069. two documents, some space groups have symmetry indices defined by band representations of inverted-space high symmetry points from which all topological invariants of the material can be derived. For a material that satisfies the compatibility relationship, the material belongs to a topologically trivial common insulator if the space group to which the material belongs has no symmetry index. And if the space group has the symmetry index, substituting the irreducible expression of the high symmetry point energy band calculated in the previous step into formulas in two documents to calculate the symmetry index of the material. If the symmetry indexes are all 0, the material is a topologically trivial ordinary insulator, if not all 0, in nsoc-setting the material belongs to a general point topology semi-metal (GMSM), in soc-setting the material belongs to a Topological Insulator (TI) or a Topological Crystal Insulator (TCI). In soc-setting, if the last strong topology index in the symmetry index is odd, the material belongs to a topological insulator, and if the last strong topology index in the symmetry index is even, the material belongs to a topological crystal insulator.
On The basis of The algorithm of The present invention, 39519 Materials with ICSD (Integrated Crystal Structure database) database numbers on The Materials Project site were calculated. All calculations are shown on the website http:// material.
The determination of the topological electronic material of the present invention is explained below by way of specific examples.
Example 1.
Tin telluride (SnTe) material of space group No. 225, is nonmagnetic, has 20 valence electrons per cell, and has 4 high symmetry points Γ ═ 0,0,0, L ═ 1/2,1/2,1/2, W ═ 1/2,1/4,3/4, X ═ 1/2,0,1/2 in brillouin zone, and the calculated energy band when considering spin-orbit coupling effect is expressed as follows:
the energy band of the SnTe material is calculated to satisfy the compatibility relation of No. 225 space group. The symmetry index of space group No. 225 is Z8, and the symmetry index of SnTe is calculated to be Z8 ═ 4, so the topological classification is "topological crystal insulator" (TCI).
Example 2.
Also SnTe, which is space group 225, when the spin-orbit coupling effect is neglected, its energy band indicates that the compatibility relationship between two highly symmetric points L and W is broken, so its topology is classified as "highly symmetric line topology half-metal" (HSLSM).
Example 3.
The material tellurite (HgTe) in space group 216 is non-magnetic and has a total of 18 valence electrons per cell. The four energy bands 17, 18, 19 and 20 of HgTe at the point gamma are found to belong to the same 4-dimensional representation through calculation
Its topology is classified as "high symmetry point topology semi-metals" (HSPSM).
The invention designs a set of rapid and full-automatic topological material searching method by using a symmetry index theory. According to the new theory, whether a material has topological properties (and what topological properties) or not can be judged, and only the symmetry of the wave function of a plurality of high-symmetry momentum points in the energy band structure needs to be analyzed (namely, irreducible representation of the wave function is calculated). The method can read the information of the high-symmetry point wave function of any crystal material from the first-nature principle calculation software, automatically complete symmetry analysis, and obtain the topological property of any material by calculating the symmetry index by using the obtained symmetry data. In the prototype design, the Vienna Ab Initio Package is selected as a first principle calculation tool to finish all codes, the method is easy to popularize to other calculation software, and only corresponding data interfaces are required to be finished.
The method greatly simplifies the calculation of the topological property of the material, and can easily carry out full-disk scanning on the existing material database to search out all topological materials on the basis of the calculation.
Although the present invention has been described by way of preferred embodiments, the present invention is not limited to the embodiments described herein, and various changes and modifications may be made without departing from the scope of the present invention.
Claims (7)
1. A method for judging a topological electronic material comprises the following steps:
step 1: judging the magnetism and the metallicity of the given material, and selecting a non-magnetic and non-metallic material;
step 2: calculating an electron wave function and an energy eigenvalue of a high symmetry point in the Brillouin zone based on the structure file of the selected material; and
and step 3: and (3) judging whether the material has degeneracy at the high-symmetry-point occupied energy band or not based on the electronic wave function and the energy eigenvalue obtained in the step (2), and if the material has degeneracy at the high-symmetry-point occupied energy band, judging that the material belongs to the high-symmetry-point topological semimetal.
2. The method for determining topological electronic material of claim 1, wherein said step 2 further comprises performing symmetry checking and structure normalization on the structure file of the selected material.
3. A method of determining a topological electronic material of claim 1, wherein if there is no degeneracy of the material at the high symmetry point occupied energy band, the following step 4 is performed:
and 4, step 4: judging whether the energy band of the material meets the compatibility relation,
wherein, if the energy band of the material does not satisfy the compatibility relationship, the material belongs to the high symmetry line topological semi-metal.
4. The method for determining topological electronic material of claim 3, wherein in step 3, a direct energy gap of the material is obtained from said energy eigenvalue, and if said direct energy gap is 2meV or more, it is determined that the material does not have degeneracy at a high symmetry point occupied energy band.
5. A method of determining a topological electronic material as claimed in claim 3, wherein in step 3 a direct energy gap of the material is obtained from said energy eigenvalues, and if said direct energy gap is less than 2meV, degenerate state calculations are performed.
6. The method for determining topological electronic material of any one of claims 3 to 5, wherein if the energy band of the material satisfies the compatibility relationship, the following step 5 is performed:
and 5: and (5) calculating a symmetry index of the material, wherein in the step 5, when the spin-orbit coupling effect is ignored, if the symmetry index is not all 0, the material is judged to belong to the 'common point topological semimetal'.
7. The method for determining a topological electronic material of claim 6, wherein in step 5, when the spin orbit coupling effect is considered, if said symmetry index is not all 0, it is determined that the material belongs to a "topological insulator" or a "topological crystal insulator".
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